t-J模型是常常用來研究高溫超導的微觀機制的模型,透過投影算符把d軌道電子雙佔據態投影掉, 因此忽略了電子在氧$2p^6$ 與$銅3d^{10}$ 之間的電荷漲落,在電荷轉移能隙大的極限是很好的近似,然而最近的掃描穿隧能譜卻發現了電荷轉移能隙只有大約1到2個電子伏特, 這樣意味著我們必須重新考慮電荷漲落在銅氧化物超導扮演的角色, 我們修改了$t-J$模型,重新引進電荷漲落的電子躍遷, 這導致了雙佔據態的產生與消失, 適當的選擇躍遷強度, 這個模型在電子跳躍強度(hopping amplitude)相同的極限下等效於 $t-J-U$模型。 我們使用了變分蒙地卡羅來研究這個模型,我們的結果也證實了配對序參量與電荷轉移能隙之間存在著反關聯, 更有趣的是, 電荷漲落對配對強度的影響, 在低參雜與過參雜表現的不一樣。 最後，我們希望可以用這個模型給出統一性的超導相圖。

$High-T_c $ Cuprates have been studied quite often as an effective one band $t - J$model that neglects charge fluctuation between oxygen $2p^6$ band and copper$3d^{10}$ band. However, recent Scanning Tunneling Spectra(STS) measurement on underdoped Cuprate shows that charge transfer gap is only of order 1-2eV. This small gap necessitates a re-examination of the charge transfer fluctuation. Here we modify the t-J model by including charge transfer fluctuation. The new model allowing the hopping that forms of doubly occupied sites(doublons) and hopping of doublon. For the same hopping amplitude it is exactly the same with the t-J-U model. This model is studied via variational Monte Carlo method(VMC). The anti-correlation between charge transfer gap and pairing is also confirmed. More interestingly the charge fluctuation is found to affect pairing order parameter in different ways in underdoped and overdoped regions.

Contents
致謝i
中文􁀇要ii
Abstract iii
Contents iv
List of Figures vi
List of Tables viii
1 Introduction 1
2 Varitional Monte Carlo Method 6
2.1 Markov Chain and Metropolis algorithm . . . . . . . . . . . . . . . . . . 7
2.2 Updating method: Sherman–Morrison formula . . . . . . . . . . . . . . 8
2.3 Trial Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Anti-symmetric configuration state function . . . . . . . . . . . . 8
2.3.2 Jastrow Projector . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Optimization Wavefunction: Stochatistic Reconfiguration(SR) . . . . . . 11
3 Results and Discussion 13
3.1 Effect of ¯t, td and ¯U . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.1 tune ¯t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.2 tune ¯U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.3 tune td . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Anti-correlation between charge transfer gap and Tc . . . . . . . . . . . . 19
3.3 Next to nearest neighbor hopping dependency and Universal Dome . . . . 20
4 Conclusion 24
Bibliography 25
Appendix 31
4.1 Hole hopping t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 charge fluctuation hopping ¯t . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 doublon hopping td . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4 Estimation of the Hopping Amplitude . . . . . . . . . . . . . . . . . . . 42

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